# Definition:Real Interval/Half-Open

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## Definition

Let $a, b \in \R$.

There are two **half-open (real) intervals** from $a$ to $b$.

#### Right half-open

The **right half-open (real) interval** from $a$ to $b$ is the subset:

- $\hointr a b := \set {x \in \R: a \le x < b}$

#### Left half-open

The **left half-open (real) interval** from $a$ to $b$ is the subset:

- $\hointl a b := \set {x \in \R: a < x \le b}$

## Also known as

This can often be seen rendered as **half open interval**.

A **half-open interval** can also be referred to as a **half-closed interval**.

## Examples

### Example $1$

Let $I$ be the unbounded closed real interval defined as:

- $I := \hointr 1 2$

Then $1 \in I$.

## Also see

- Definition:Open Real Interval
- Definition:Closed Real Interval
- Definition:Unbounded Open Real Interval
- Definition:Unbounded Closed Real Interval

- Definition:Half-Open Rectangle, a generalization to higher dimensional spaces

## Sources

- 1970: Arne Broman:
*Introduction to Partial Differential Equations*... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.1$ Definitions - 1991: Felix Hausdorff:
*Set Theory*(4th ed.) ... (previous) ... (next): Preliminary Remarks - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**open interval** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**open interval**

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- 1963: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion